detemináns

The determinant, a scalar value associated with a square matrix, is a fundamental concept in linear algebra. It summarizes key properties of the linear transformation represented by the matrix, such as invertibility and how volumes are scaled under the transformation. In some languages, the term is written determináns, and the form detemináns may appear as a variant spelling.

For a 2×2 matrix, the determinant is straightforward: det([ [a, b], [c, d] ]) = ad − bc. For larger matrices, determinants can be computed by expanding along a row or column, or more efficiently through methods such as LU decomposition or using a matrix’s triangular form.

Key properties include: det(AB) = det(A) det(B); det(A^T) = det(A); det(kA) = k^n det(A) for an n×n matrix; det(A) = 0 if the rows or columns are linearly dependent; det(A) ≠ 0 implies A is invertible, and det(A⁻¹) = 1/det(A). Row operations affect determinants in specific ways: swapping two rows multiplies the determinant by −1, while adding a multiple of one row to another row does not change it. For triangular matrices, the determinant is the product of the diagonal entries; the identity matrix has determinant 1.

Applications of determinants include solving linear systems (Cramer’s rule applies when det ≠ 0), testing invertibility, and understanding geometric effects: the determinant measures the volume (or area) scaling of the linear transformation. In calculus, the determinant appears as the Jacobian determinant in change of variables for integrals.